Poincaré Map

 A continuous flow can generate a discrete map in at least two ways: by a time-T map and by a Poincaré map. A time-T map results when a flow is sampled at a fixed time interval T. That is, the flow is sampled whenever t = nT for n = 0, 1, 2, 3, and so on.

The more important way (as described, for instance, in Guckenheimer and Holmes [1]) in which a continuous flow generates a discrete map is via a Poincaré map. Let  be an orbit of a flow in  . As illustrated in Figure 4.2, it is often possible to find a local cross section  about  , which is of dimension n-1.

The cross section need not be planar; however, it must be transverse to the flow. All the orbits in the neighborhood of  must pass through  . The technical requirement is that for all  , where  is the unit normal vector to  at  . Let  be a point where  intersects  , and let  be a point in the neighborhood of  . Then the Poincaré map (or first return map) is defined by

 

where  is the time taken for an orbit starting at  to return to  . It is useful to define a Poincaré map in the neighborhood of a periodic orbit. If the orbit  is periodic of period T, then  . A periodic orbit that returns directly to itself is a fixed point of the Poincaré map. Moreover, an orbit starting at q close to p will have a return time close to T.

For forced systems, such as the Duffing oscillator studied in section 3.4, a global cross section and Poincaré map are easy to define since the phase space topology is  . All periodic orbits of a forced system have a period that is an integer multiple of the forcing period. In this situation, it is sensible to pick a planar global cross section that is transverse to  (see Fig. 3.8). The return time for this cross section is independent of position and equals the forcing period. In this special case, the Poincaré map is equivalent to a time-T map.

The Poincaré map for the example of the rotational flow generated by  is particularly trivial. A good cross section is defined by the positive half of the x-axis,  . All the orbits of the flow are fixed points in the cross section, so the Poincaré map is just the identity map, P(x) = x.

See Appendix E, Hénon’s Trick, for a discussion of the numerical calculation of a Poincaré map from a cross section.

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